@article{COIFMAN20065,
title = {Diffusion maps},
journal = {Applied and Computational Harmonic Analysis},
volume = {21},
number = {1},
pages = {5-30},
year = {2006},
note = {Special Issue: Diffusion Maps and Wavelets},
issn = {1063-5203},
doi = {https://doi.org/10.1016/j.acha.2006.04.006},
url = {https://www.sciencedirect.com/science/article/pii/S1063520306000546},
author = {Ronald R. Coifman and Stéphane Lafon},
keywords = {Diffusion processes, Diffusion metric, Manifold learning, Dimensionality reduction, Eigenmaps, Graph Laplacian},
abstract = {In this paper, we provide a framework based upon diffusion processes for finding meaningful geometric descriptions of data sets. We show that eigenfunctions of Markov matrices can be used to construct coordinates called diffusion maps that generate efficient representations of complex geometric structures. The associated family of diffusion distances, obtained by iterating the Markov matrix, defines multiscale geometries that prove to be useful in the context of data parametrization and dimensionality reduction. The proposed framework relates the spectral properties of Markov processes to their geometric counterparts and it unifies ideas arising in a variety of contexts such as machine learning, spectral graph theory and eigenmap methods.}
}

@inbook{POD_1,
author = {Julien Weiss},
title = {A Tutorial on the Proper Orthogonal Decomposition},
booktitle = {AIAA Aviation 2019 Forum},
publisher = {American Institute of Aeronautics and Astronautics},
chapter = {},
pages = {},
year = {2019},
doi = {10.2514/6.2019-3333},
URL = {https://arc.aiaa.org/doi/abs/10.2514/6.2019-3333},
eprint = {https://arc.aiaa.org/doi/pdf/10.2514/6.2019-3333}
}

@article{POD_2,
  doi = {10.1090/qam/910462},
  url = {https://doi.org/10.1090/qam/910462},
  year = {1987},
  journal = {Quarterly of applied Mathematics},
  publisher = {American Mathematical Society ({AMS})},
  volume = {45},
  number = {3},
  pages = {561--571},
  author = {Lawrence Sirovich},
  title = {Turbulence and the dynamics of coherent structures. I. Coherent structures}
}

@article{LARS,
	author = {Bradley Efron and Trevor Hastie and Iain Johnstone and Robert Tibshirani},
	title = {Least angle regression},
	journal = {The Annals of Statistics},
	year = {2004},
	volume = {32},
	number = {2},
  ISSN = {00905364},
  doi = {10.2307/3448465},
  pages = {407--451},
  publisher = {Institute of Mathematical Statistics},
}

@article{BLATMANLARS,
	author = {G{\'{e}}raud Blatman and Bruno Sudret},
	title = {Adaptive sparse polynomial chaos expansion based on least angle regression},
	journal = {Journal of Computational Physics},
	year = {2011},
	volume = {230},
	number = {6},
   pages = {2345--2367},
	doi = {10.1016/j.jcp.2010.12.021},
}

@article{HOSVD_1,
author = {Giovanis, D.G. and Shields, M.D.},
title = {Variance-based simplex stochastic collocation with model order reduction for high-dimensional systems},
journal = {International Journal for Numerical Methods in Engineering},
volume = {117},
number = {11},
pages = {1079-1116},
keywords = {higher-order singular value decomposition, multi-element collocation, polynomial chaos, reduced-order model, simplex stochastic collocation, singular value decomposition},
doi = {https://doi.org/10.1002/nme.5992},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5992},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5992},
abstract = {Summary In this work, an adaptive simplex stochastic collocation method is introduced in which sample refinement is informed by variability in the solution of the system. The proposed method is based on the concept of multi-element stochastic collocation methods and is capable of dealing with very high-dimensional models whose solutions are expressed as a vector, a matrix, or a tensor. The method leverages random samples to create a multi-element polynomial chaos surrogate model that incorporates local anisotropy in the refinement, informed by the variance of the estimated solution. This feature makes it beneficial for strongly nonlinear and/or discontinuous problems with correlated non-Gaussian uncertainties. To solve large systems, a reduced-order model (ROM) of the high-dimensional response is identified using singular value decomposition (higher-order SVD for matrix/tensor solutions) and polynomial chaos is used to interpolate the ROM. The method is applied to several stochastic systems of varying type of response (scalar/vector/matrix) and it shows considerable improvement in performance compared to existing simplex stochastic collocation methods and adaptive sparse grid collocation methods.},
year = {2019}
}

@article{HOSVD_2,
  doi = {10.1137/s0895479896305696},
  url = {https://doi.org/10.1137/s0895479896305696},
  year = {2000},
  month = jan,
  journal = {SIAM Journal on Matrix Analysis and Applications},
  publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
  volume = {21},
  number = {4},
  pages = {1253--1278},
  author = {Lieven De Lathauwer and Bart De Moor and Joos Vandewalle},
  title = {A Multilinear Singular Value Decomposition}
}

@misc{Grassmann_1,
      title={A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects},
      author={Thomas Bendokat and Ralf Zimmermann and P. -A. Absil},
      year={2020},
      eprint={2011.13699},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}

@article{Grassmann_2,
  doi = {10.1023/b:acap.0000013855.14971.91},
  url = {https://doi.org/10.1023/b:acap.0000013855.14971.91},
  year = {2004},
  month = jan,
  journal = {Acta Applicandae Mathematica},
  publisher = {Springer Science and Business Media {LLC}},
  volume = {80},
  number = {2},
  pages = {199--220},
  author = {P.-A. Absil and R. Mahony and R. Sepulchre},
  title = {Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation}
}

@Book{Distances_1,
  Title                    = {Matrix Computations},
  Author                   = {Golub, Gene H. and Van Loan, Charles F.},
  Publisher                = {The Johns Hopkins University Press},
  Year                     = {1996},
  Edition                  = {Third}
}

@misc{Distances_2,
      title={Schubert varieties and distances between subspaces of different dimensions},
      author={Ke Ye and Lek-Heng Lim},
      year={2016},
      eprint={1407.0900},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}

@inproceedings{kernels_1,
author = {Hamm, Jihun and Lee, Daniel D.},
title = {Grassmann Discriminant Analysis: A Unifying View on Subspace-Based Learning},
year = {2008},
isbn = {9781605582054},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/1390156.1390204},
doi = {10.1145/1390156.1390204},
booktitle = {Proceedings of the 25th International Conference on Machine Learning},
pages = {376–383},
numpages = {8},
location = {Helsinki, Finland},
series = {ICML '08}
}

@misc{kernels_2,
      title={Expanding the Family of Grassmannian Kernels: An Embedding Perspective},
      author={Mehrtash T. Harandi and Mathieu Salzmann and Sadeep Jayasumana and Richard Hartley and Hongdong Li},
      year={2014},
      eprint={1407.1123},
      archivePrefix={arXiv},
      primaryClass={cs.CV}
}

@book{GNU_parallel,
      title={GNU Parallel 2018},
      ISBN={978-1-387-50988-1},
      DOI={10.5281/zenodo.1146014},
      journal={GNU Parallel 2018},
      publisher={Ole Tange},
      author={Tange, Ole},
      year={2018},
      month={Apr},
      pages={112}
}

@article{UQpy_paper,
title = {UQpy: A general purpose Python package and development environment for uncertainty quantification},
journal = {Journal of Computational Science},
volume = {47},
pages = {101204},
year = {2020},
issn = {1877-7503},
doi = {https://doi.org/10.1016/j.jocs.2020.101204},
url = {https://www.sciencedirect.com/science/article/pii/S1877750320305056},
author = {Audrey Olivier and Dimitris G. Giovanis and B.S. Aakash and Mohit Chauhan and Lohit Vandanapu and Michael D. Shields},
keywords = {Uncertainty quantification, Computational modeling, High-performance computing, Python, Software},
}

@ARTICLE{Scipy_paper,
  author  = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and
            Haberland, Matt and Reddy, Tyler and Cournapeau, David and
            Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and
            Bright, Jonathan and {van der Walt}, St{\'e}fan J. and
            Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and
            Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and
            Kern, Robert and Larson, Eric and Carey, C J and
            Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and
            {VanderPlas}, Jake and Laxalde, Denis and Perktold, Josef and
            Cimrman, Robert and Henriksen, Ian and Quintero, E. A. and
            Harris, Charles R. and Archibald, Anne M. and
            Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and
            {van Mulbregt}, Paul and {SciPy 1.0 Contributors}},
  title   = {{{SciPy} 1.0: Fundamental Algorithms for Scientific
            Computing in Python}},
  journal = {Nature Methods},
  year    = {2020},
  volume  = {17},
  pages   = {261--272},
  adsurl  = {https://rdcu.be/b08Wh},
  doi     = {10.1038/s41592-019-0686-2},
}

@book{InfoModelSelection,
  doi = {10.1007/b97636},
  url = {https://doi.org/10.1007/b97636},
  year = {2004},
  publisher = {Springer New York},
  editor = {Kenneth P. Burnham and David R. Anderson},
  title = {Model Selection and Multimodel Inference}
}

@article{BayesModelSelection,
author = {Raftery, Adrian and Newton, Michael and Satagopan, Jaya and Krivitsky, Pavel},
year = {2007},
month = {01},
pages = {},
title = {Estimating the integrated likelihood via posterior simulation using the harmonic mean identity},
volume = {8},
journal = {Bayesian statistics}
}

@article{SubsetSimulation,
title = {Estimation of small failure probabilities in high dimensions by subset simulation},
journal = {Probabilistic Engineering Mechanics},
volume = {16},
number = {4},
pages = {263-277},
year = {2001},
issn = {0266-8920},
doi = {https://doi.org/10.1016/S0266-8920(01)00019-4},
url = {https://www.sciencedirect.com/science/article/pii/S0266892001000194},
author = {Siu-Kui Au and James L. Beck},
keywords = {Markov chain Monte Carlo method, Monte Carlo simulation, Reliability, First excursion probability, First passage problem, Metropolis algorithm},
}

@article{TaylorSeries1,
title = {Structural reliability under combined random load sequences},
journal = {Computers & Structures},
volume = {9},
number = {5},
pages = {489-494},
year = {1978},
issn = {0045-7949},
doi = {https://doi.org/10.1016/0045-7949(78)90046-9},
url = {https://www.sciencedirect.com/science/article/pii/0045794978900469},
author = {Rüdiger Rackwitz and Bernd Flessler},
abstract = {An algorithm for the calculation of structural reliability under combined loading is formulated. Loads or any other actions upon structures are modelled as independent random sequences. The relevant limit state criterion is pointwise approximated by a tangent hyperplane. The combination of time-variant actions then reduces to the calculation of the maximum of a sum of random variables which is facilitated through proper, discrete approximation of extreme value and other non-normal distribution functions by normal distributions. The iteration algorithm searches for an approximation point on the limit state criterion where the probability content of the failure domain limited by the tangent hyperplane reaches its maximum. Any type of continuous limit state criterion and any distribution type for the loads can be dealt with. The method is illustrated for a section of a wall without tensile strength loaded by a bending moment and a normal force.}
}

@article{TaylorSeries2,
title = {Asymptotic approximations for multivariate integrals with an application to multinormal probabilities},
journal = {Journal of Multivariate Analysis},
volume = {30},
number = {1},
pages = {80-97},
year = {1989},
issn = {0047-259X},
doi = {https://doi.org/10.1016/0047-259X(89)90089-4},
url = {https://www.sciencedirect.com/science/article/pii/0047259X89900894},
author = {K. Breitung and M. Hohenbichler},
keywords = {multinormal distribution, asymptotic expansions, Laplace method, Mill's ratio},
abstract = {The efficient computation of multinormal integrals is an important problem of multivariate statistics. In this paper it is shown, that using methods of asymptotic analysis, asymptotic expansions for multinormal integrals can be obtained. These results are an extension of a result obtained by Ruben (1964, J. Res. Nat. Bur. Standards B68, No. 1 3–11). While the approximations of Ruben are valid only for domains bounded by hyperplanes, the results given here also apply to domains with nonlinear boundaries.}
}

@book{StratifiedSampling1,
  title={The Art of Simulation},
  author={Tocher, K.D.},
  lccn={64009760},
  series={Electrical engineering series},
  url={https://books.google.com/books?id=ccEvAAAAYAAJ},
  year={1963},
  publisher={English Universities Press}
}

@article{Rss1,
title = {Refined Stratified Sampling for efficient Monte Carlo based uncertainty quantification},
journal = {Reliability Engineering & System Safety},
volume = {142},
pages = {310-325},
year = {2015},
issn = {0951-8320},
doi = {https://doi.org/10.1016/j.ress.2015.05.023},
url = {https://www.sciencedirect.com/science/article/pii/S0951832015001726},
author = {Michael D. Shields and Kirubel Teferra and Adam Hapij and Raymond P. Daddazio},
keywords = {Uncertainty quantification, Monte Carlo simulation, Stratified sampling, Latin hypercube sampling, Sample size extension},
abstract = {A general adaptive approach rooted in stratified sampling (SS) is proposed for sample-based uncertainty quantification (UQ). To motivate its use in this context the space-filling, orthogonality, and projective properties of SS are compared with simple random sampling and Latin hypercube sampling (LHS). SS is demonstrated to provide attractive properties for certain classes of problems. The proposed approach, Refined Stratified Sampling (RSS), capitalizes on these properties through an adaptive process that adds samples sequentially by dividing the existing subspaces of a stratified design. RSS is proven to reduce variance compared to traditional stratified sample extension methods while providing comparable or enhanced variance reduction when compared to sample size extension methods for LHS – which do not afford the same degree of flexibility to facilitate a truly adaptive UQ process. An initial investigation of optimal stratification is presented and motivates the potential for major advances in variance reduction through optimally designed RSS. Potential paths for extension of the method to high dimension are discussed. Two examples are provided. The first involves UQ for a low dimensional function where convergence is evaluated analytically. The second presents a study to asses the response variability of a floating structure to an underwater shock.}
}

@article{Rss2,
title = {Adaptive Monte Carlo analysis for strongly nonlinear stochastic systems},
journal = {Reliability Engineering & System Safety},
volume = {175},
pages = {207-224},
year = {2018},
issn = {0951-8320},
doi = {https://doi.org/10.1016/j.ress.2018.03.018},
url = {https://www.sciencedirect.com/science/article/pii/S0951832017308827},
author = {Michael D. Shields},
keywords = {Uncertainty quantification, Monte Carlo simulation, Stratified sampling, Latin hypercube sampling, Non-linear systems, Stochastic systems, Importance sampling},
abstract = {This paper compares space-filling and importance sampling (IS)-based Monte Carlo sample designs with those derived for optimality in the error of stratified statistical estimators. Space-filling designs are shown to be optimal for systems whose response depends linearly on the input random variables. They are, however, shown to be far from optimal when the system is nonlinear. To achieve optimality, it is shown that samples should be placed densely in regions of large variation (sparsely in regions of small variation). This notion is shown to be subtly, but importantly, different from other non-space-filling designs, particularly IS. To achieve near-optimal sample designs, the adaptive Gradient Enhanced Refined Stratified Sampling (GE-RSS) is proposed that sequentially refines the probability space in accordance with stratified sampling. The space is refined according to the estimated local variance of the system computed from gradients using a surrogate model. The method significantly reduces the error in stratified Monte Carlo estimators for strongly nonlinear systems, outperforms both space-filling methods and IS-based methods, and is simple to implement. Numerical examples on strongly nonlinear systems illustrate the improvement over space-filling and IS designs. The method is applied to study the probability of shear band formation in a bulk metallic glass.}
}

@article{Simplex1,
author = {Edeling, Wouter and Dwight, Richard and Cinnella, Paola},
year = {2016},
month = {04},
pages = {301-328},
title = {Simplex-Stochastic Collocation Method With Improved Scalability},
volume = {310},
journal = {Journal of Computational Physics},
doi = {10.1016/j.jcp.2015.12.034}
}

@article{AKMCS1,
title = {AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation},
journal = {Structural Safety},
volume = {33},
number = {2},
pages = {145-154},
year = {2011},
issn = {0167-4730},
doi = {https://doi.org/10.1016/j.strusafe.2011.01.002},
url = {https://www.sciencedirect.com/science/article/pii/S0167473011000038},
author = {B. Echard and N. Gayton and M. Lemaire},
keywords = {Reliability, Metamodel, Kriging, Active learning, Monte Carlo, Failure probability},
abstract = {An important challenge in structural reliability is to keep to a minimum the number of calls to the numerical models. Engineering problems involve more and more complex computer codes and the evaluation of the probability of failure may require very time-consuming computations. Metamodels are used to reduce these computation times. To assess reliability, the most popular approach remains the numerous variants of response surfaces. Polynomial Chaos [1] and Support Vector Machine [2] are also possibilities and have gained considerations among researchers in the last decades. However, recently, Kriging, originated from geostatistics, have emerged in reliability analysis. Widespread in optimisation, Kriging has just started to appear in uncertainty propagation [3] and reliability [4], [5] studies. It presents interesting characteristics such as exact interpolation and a local index of uncertainty on the prediction which can be used in active learning methods. The aim of this paper is to propose an iterative approach based on Monte Carlo Simulation and Kriging metamodel to assess the reliability of structures in a more efficient way. The method is called AK-MCS for Active learning reliability method combining Kriging and Monte Carlo Simulation. It is shown to be very efficient as the probability of failure obtained with AK-MCS is very accurate and this, for only a small number of calls to the performance function. Several examples from literature are performed to illustrate the methodology and to prove its efficiency particularly for problems dealing with high non-linearity, non-differentiability, non-convex and non-connex domains of failure and high dimensionality.}
}


@article{AKMCS2,
  title = {Efficient global optimization of expensive black-box functions.” Journal of Global optimization},
  doi = {10.1023/a:1008306431147},
  url = {https://doi.org/10.1023/a:1008306431147},
  year = {1998},
  publisher = {Springer Science and Business Media {LLC}},
  volume = {13},
  number = {4},
  pages = {455--492},
  author = {Donald R. Jones and Matthias Schonlau and William J. Welch},
  journal = {Journal of Global Optimization}
}

@article{AKMCS3,
    author = {V. S. Sundar  and Michael D. Shields },
    title = {Reliability Analysis Using Adaptive Kriging Surrogates with Multimodel Inference},
    journal = {ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering},
    volume = {5},
    number = {2},
    year = {2019},
    doi = {10.1061/AJRUA6.0001005}
}

@article{AKMCS4,
author = {Bichon, B. J. and Eldred, M. S. and Swiler, L. P. and Mahadevan, S. and McFarland, J. M.},
title = {Efficient Global Reliability Analysis for Nonlinear Implicit Performance Functions},
journal = {AIAA Journal},
volume = {46},
number = {10},
pages = {2459-2468},
year = {2008},
doi = {10.2514/1.34321},
}

@book{AKMCS5,
author = {Lam, Chen Quin},
advisor = {Notz, William},
title = {Sequential Adaptive Designs in Computer Experiments for Response Surface Model Fit},
year = {2008},
isbn = {9780549716860},
publisher = {Ohio State University},
address = {USA},
note = {AAI3321369},
}

@book{MCMC1,
  doi = {10.1201/b16018},
  url = {https://doi.org/10.1201/b16018},
  year = {2013},
  month = nov,
  publisher = {Chapman and Hall/{CRC}},
  author = {Andrew Gelman and John B. Carlin and Hal S. Stern and David B. Dunson and Aki Vehtari and Donald B. Rubin},
  title = {Bayesian Data Analysis}
}

@Book{MCMC2,
 author = {Smith, Ralph},
 title = {Uncertainty quantification : Theory, Implementation, and Applications},
 publisher = {Society for Industrial and Applied Mathematics},
 year = {2013},
 address = {Philadelphia},
 isbn = {978-1-611973-21-1}
 }

@article{Dram1,
  doi = {10.1007/s11222-006-9438-0},
  url = {https://doi.org/10.1007/s11222-006-9438-0},
  year = {2006},
  month = dec,
  publisher = {Springer Science and Business Media {LLC}},
  volume = {16},
  number = {4},
  pages = {339--354},
  author = {Heikki Haario and Marko Laine and Antonietta Mira and Eero Saksman},
  title = {{DRAM}: Efficient adaptive {MCMC}},
  journal = {Statistics and Computing}
}


@article{Dream1,
title = {Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling},
author = {Vrugt, Jasper A and Hyman, James M and Robinson, Bruce A and Higdon, Dave and Ter Braak, Cajo J F and Diks, Cees G H},
url = {https://www.osti.gov/biblio/960766}, journal = {International Journal of Nonlinear Sciences and Numerical Simulation},
place = {United States},
year = {2008},
month = {1}
}

@article{Dream2,
title = {Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation},
journal = {Environmental Modelling & Software},
volume = {75},
pages = {273-316},
year = {2016},
issn = {1364-8152},
doi = {https://doi.org/10.1016/j.envsoft.2015.08.013},
url = {https://www.sciencedirect.com/science/article/pii/S1364815215300396},
author = {Jasper A. Vrugt},
keywords = {Bayesian inference, Markov chain Monte Carlo (MCMC) simulation, Random walk metropolis (RWM), Adaptive metropolis (AM), Differential evolution Markov chain (DE-MC), Prior distribution, Likelihood function, Posterior distribution, Approximate Bayesian computation (ABC), Diagnostic model evaluation, Residual analysis, Environmental modeling, Bayesian model averaging (BMA), Generalized likelihood uncertainty estimation (GLUE), Multi-processor computing, Extended metropolis algorithm (EMA)},
abstract = {Bayesian inference has found widespread application and use in science and engineering to reconcile Earth system models with data, including prediction in space (interpolation), prediction in time (forecasting), assimilation of observations and deterministic/stochastic model output, and inference of the model parameters. Bayes theorem states that the posterior probability, p(H|Y˜) of a hypothesis, H is proportional to the product of the prior probability, p(H) of this hypothesis and the likelihood, L(H|Y˜) of the same hypothesis given the new observations, Y˜, or p(H|Y˜)∝p(H)L(H|Y˜). In science and engineering, H often constitutes some numerical model, ℱ(x) which summarizes, in algebraic and differential equations, state variables and fluxes, all knowledge of the system of interest, and the unknown parameter values, x are subject to inference using the data Y˜. Unfortunately, for complex system models the posterior distribution is often high dimensional and analytically intractable, and sampling methods are required to approximate the target. In this paper I review the basic theory of Markov chain Monte Carlo (MCMC) simulation and introduce a MATLAB toolbox of the DiffeRential Evolution Adaptive Metropolis (DREAM) algorithm developed by Vrugt et al. (2008a, 2009a) and used for Bayesian inference in fields ranging from physics, chemistry and engineering, to ecology, hydrology, and geophysics. This MATLAB toolbox provides scientists and engineers with an arsenal of options and utilities to solve posterior sampling problems involving (among others) bimodality, high-dimensionality, summary statistics, bounded parameter spaces, dynamic simulation models, formal/informal likelihood functions (GLUE), diagnostic model evaluation, data assimilation, Bayesian model averaging, distributed computation, and informative/noninformative prior distributions. The DREAM toolbox supports parallel computing and includes tools for convergence analysis of the sampled chain trajectories and post-processing of the results. Seven different case studies illustrate the main capabilities and functionalities of the MATLAB toolbox.}
}

@article{Stretch1,
  doi = {10.2140/camcos.2010.5.65},
  url = {https://doi.org/10.2140/camcos.2010.5.65},
  year = {2010},
  month = jan,
  publisher = {Mathematical Sciences Publishers},
  volume = {5},
  number = {1},
  pages = {65--80},
  author = {Jonathan Goodman and Jonathan Weare},
  title = {Ensemble samplers with affine invariance},
  journal = {Communications in Applied Mathematics and Computational Science}
}

@article{Stretch2,
   title={emcee: The MCMC Hammer},
   volume={125},
   ISSN={1538-3873},
   url={http://dx.doi.org/10.1086/670067},
   DOI={10.1086/670067},
   number={925},
   journal={Publications of the Astronomical Society of the Pacific},
   publisher={IOP Publishing},
   author={Foreman-Mackey, Daniel and Hogg, David W. and Lang, Dustin and Goodman, Jonathan},
   year={2013},
   month={Mar},
   pages={306–312}
}

@article{StochasticProcess1,
title = {Digital simulation of random processes and its applications},
journal = {Journal of Sound and Vibration},
volume = {25},
number = {1},
pages = {111-128},
year = {1972},
issn = {0022-460X},
doi = {https://doi.org/10.1016/0022-460X(72)90600-1},
url = {https://www.sciencedirect.com/science/article/pii/0022460X72906001},
author = {M. Shinozuka and C.-M. Jan},
abstract = {Efficient methods are presented for digital simulation of a general homogeneous process (multidimensional or multivariate or multivariate-multidimensional) as a series of cosine functions with weighted amplitudes, almost evenly spaced frequencies, and random phase angles. The approach is also extended to the simulation of a general non-homogeneous oscillatory process characterized by an evolutionary power spectrum. Generalized forces involved in the modal analysis of linear or non-linear structures can be efficiently simulated as a multivariate process using the cross-spectral density matrix computed from the spectral density function of the multidimensional excitation process. Possible applications include simulation of (i) wind-induced ocean wave elevation, (ii) spatial random variation of material properties, (iii) the fluctuating part of atmospheric wind velocities and (iv) random surface roughness of highways and airport runways.}
}

@article{StochasticProcess2,
    author = {Shinozuka, Masanobu and Deodatis, George},
    title = "{Simulation of Stochastic Processes by Spectral Representation}",
    journal = {Applied Mechanics Reviews},
    volume = {44},
    number = {4},
    pages = {191-204},
    year = {1991},
    month = {04},
    abstract = "{The subject of this paper is the simulation of one-dimensional, uni-variate, stationary, Gaussian stochastic processes using the spectral representation method. Following this methodology, sample functions of the stochastic process can be generated with great computational efficiency using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number N of the terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the temporally-averaged mean value and the autocorrelation function are identical with the corresponding targets, when the averaging takes place over the fundamental period of the cosine series. The most important property of the simulated stochastic process is that it is asymptotically Gaussian as N → ∞. Another attractive feature of the method is that the cosine series formula can be numerically computed efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in engineering mechanics and structural engineering. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response variability due to system stochasticity).}",
    issn = {0003-6900},
    doi = {10.1115/1.3119501},
    url = {https://doi.org/10.1115/1.3119501},
    eprint = {https://asmedigitalcollection.asme.org/appliedmechanicsreviews/article-pdf/44/4/191/5435905/191\_1.pdf},
}

@article{StochasticProcess3,
    author = {Shinozuka, Masanobu and Deodatis, George},
    title = "{Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation}",
    journal = {Applied Mechanics Reviews},
    volume = {49},
    number = {1},
    pages = {29-53},
    year = {1996},
    month = {01},
    abstract = "{The subject of this paper is the simulation of multi-dimensional, homogeneous, Gaussian stochastic fields using the spectral representation method. Following this methodology, sample functions of the stochastic field can be generated using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic field when the number of terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the spatially-averaged mean value, autocorrelation function and power spectral density function are identical with the corresponding targets, when the averaging takes place over the multi-dimensional domain associated with the fundamental period of the cosine series. Another property of the simulated stochastic field is that it is asymptotically Gaussian as the number of terms in the cosine series approaches infinity. The most important feature of the method is that the cosine series formula can be numerically computed very efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in structural engineering, engineering mechanics and physics. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response variability due to system stochasticity).}",
    issn = {0003-6900},
    doi = {10.1115/1.3101883},
    url = {https://doi.org/10.1115/1.3101883},
    eprint = {https://asmedigitalcollection.asme.org/appliedmechanicsreviews/article-pdf/49/1/29/5437403/29\_1.pdf},
}


@article{StochasticProcess4,
author = {George Deodatis },
title = {Simulation of Ergodic Multivariate Stochastic Processes },
journal = {Journal of Engineering Mechanics},
volume = {122},
number = {8},
pages = {778-787},
year = {1996},
doi = {10.1061/(ASCE)0733-9399(1996)122:8(778)}
}

@article{StochasticProcess5,
author = {Huang, S. P. and Quek, S. T. and Phoon, K. K.},
title = {Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes},
journal = {International Journal for Numerical Methods in Engineering},
volume = {52},
number = {9},
pages = {1029-1043},
keywords = {Karhunen–Loeve expansion, stationary Gaussian process, non-stationary Gaussian process, stochastic series representation, simulation, covariance models},
doi = {https://doi.org/10.1002/nme.255},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.255},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.255},
abstract = {Abstract A random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen-decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non-stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second-order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen-solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non-stationary processes with little additional effort. Copyright © 2001 John Wiley \& Sons, Ltd.},
year = {2001}
}

@article{StochasticProcess6,
title = {Simulation of second-order processes using Karhunen–Loeve expansion},
journal = {Computers & Structures},
volume = {80},
number = {12},
pages = {1049-1060},
year = {2002},
issn = {0045-7949},
doi = {https://doi.org/10.1016/S0045-7949(02)00064-0},
url = {https://www.sciencedirect.com/science/article/pii/S0045794902000640},
author = {K.K. Phoon and S.P. Huang and S.T. Quek},
keywords = {Simulation, Karhunen–Loeve expansion, Non-Gaussian process, Non-stationary process, Target covariance function, Target marginal distribution function},
abstract = {A unified and practical framework is developed for generating second-order stationary and non-stationary, Gaussian and non-Gaussian processes with a specified marginal distribution function and covariance function. It utilizes the Karhunen–Loeve expansion for simulation and an iterative mapping scheme to fit the target marginal distribution function. The proposed method has three main advantages: (a) processes with Gaussian-like marginal distribution can be generated almost directly without iteration, (b) distributions that deviate significantly from the Gaussian case can be handled efficiently and (c) non-stationary processes can be generated within the same unified framework. Four numerical examples are used to demonstrate the validity and convergence characteristics of the proposed algorithm. Based on these examples, it was shown that the proposed algorithm is more robust and general than the commonly used spectral representation method.}
}

@article{StochasticProcess7,
title = {Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {271},
pages = {109-129},
year = {2014},
issn = {0045-7825},
doi = {https://doi.org/10.1016/j.cma.2013.12.010},
url = {https://www.sciencedirect.com/science/article/pii/S0045782513003502},
author = {Wolfgang Betz and Iason Papaioannou and Daniel Straub},
keywords = {Random field discretization, Karhunen–Loève expansion, Nyström method, Collocation method, Galerkin method, Finite cell method},
abstract = {The computational efficiency of random field representations with the Karhunen–Loève (KL) expansion relies on the solution of a Fredholm integral eigenvalue problem. This contribution compares different methods that solve this problem. Focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions. These include the Nyström method as well as collocation and Galerkin projection methods. Among the Galerkin methods, we investigate the finite element method (FEM) and propose the application of the finite cell method (FCM). This method is based on an extension to the FEM but avoids mesh generation on domains of complex geometric shape. The FCM was originally presented in Parvizian et al. (2007) [17] for the solution of elliptic boundary value problems. As an alternative to the L2-projection of the covariance function used in the Galerkin method, H1/2-projection and discrete projection are investigated. It is shown that the expansion optimal linear estimation (EOLE) method proposed in Li and Der Kiureghian (1993) [18] constitutes a special case of the Nyström method. It is found that the EOLE method is most efficient for the numerical solution of the KL expansion. The FEM and the FCM are more efficient than the EOLE method in evaluating a realization of the random field and, therefore, are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime – e.g., in finite element reliability analysis.}
}

@article{StochasticProcess8,
  title={Simulation of higher-order stochastic processes by spectral representation},
  author={Michael D. Shields and Hwanpyo Kim},
  journal={Probabilistic Engineering Mechanics},
  year={2017},
  volume={47},
  pages={1-15}
}

@article{StochasticProcess9,
title = {3rd-order Spectral Representation Method: Simulation of multi-dimensional random fields and ergodic multi-variate random processes with fast Fourier transform implementation},
journal = {Probabilistic Engineering Mechanics},
volume = {64},
pages = {103128},
year = {2021},
issn = {0266-8920},
doi = {https://doi.org/10.1016/j.probengmech.2021.103128},
url = {https://www.sciencedirect.com/science/article/pii/S0266892021000126},
author = {Lohit Vandanapu and Michael D. Shields},
keywords = {Stochastic fields, Random fields, Spectral representation, Fast Fourier transform, Simulation},
}

@misc{StochasticProcess10,
      title={3rd-order Spectral Representation Method: Part II -- Ergodic Multi-variate random processes with fast Fourier transform},
      author={Lohit Vandanapu and Michael D. Shields},
      year={2019},
      eprint={1911.10251},
      archivePrefix={arXiv},
      primaryClass={math.ST}
}

@inproceedings{StochasticProcess11,
  title={Applied non-Gaussian processes : examples, theory, simulation, linear random vibration, and MATLAB solutions},
  booktitle={Applied non-Gaussian processes : examples, theory, simulation, linear random vibration, and MATLAB solutions},
  author={Mircea Grigoriu},
  year={1995}
}

@article{StochasticProcess12,
title = {A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic vector process by a translation process with applications in wind velocity simulation},
journal = {Probabilistic Engineering Mechanics},
volume = {31},
pages = {19-29},
year = {2013},
issn = {0266-8920},
doi = {https://doi.org/10.1016/j.probengmech.2012.10.003},
url = {https://www.sciencedirect.com/science/article/pii/S0266892012000549},
author = {M.D. Shields and G. Deodatis},
keywords = {Non-Gaussian stochastic processes, Translation processes, Simulation, Stochastic vector processes, Wind velocity simulation, Wind pressure simulation},
abstract = {Several methodologies utilize translation vector process theory for simulation of non-Gaussian stochastic vector processes and fields. However, translation theory imposes certain compatibility conditions on the non-Gaussian cross-spectral density matrix (CSDM) and the non-Gaussian marginal probability density functions (PDFs). For many practical applications such as simulation of wind velocity time histories, the non-Gaussian CSDM and PDFs are assigned arbitrarily. As a result, they are often incompatible. The generally accepted approach to addressing this incompatibility is to approximate the incompatible pair of CSDM/PDFs with a compatible pair that closely matches the incompatible pair. A limited number of techniques are available to do so and these methodologies are usually complicated and time consuming. In this paper, a novel iterative methodology is presented that simply and efficiently estimates a non-Gaussian CSDM that: (a) is compatible with the prescribed non-Gaussian PDFs and (b) closely approximates the prescribed incompatible non-Gaussian CSDM. The corresponding underlying Gaussian CSDM is also determined and used for simulation purposes. Numerical examples are provided demonstrating the capabilities of the methodology for both general non-Gaussian stochastic vector processes and a non-Gaussian vector wind velocity process.}
}

@article{StochasticProcess13,
author = {Kim, Hwanpyo and Shields, Michael},
year = {2015},
month = {08},
pages = {31-42},
title = {Modeling strongly non-Gaussian non-stationary stochastic processes using the Iterative Translation Approximation Method and Karhunen-Loeve expansion},
volume = {161},
journal = {Computers & Structures},
doi = {10.1016/j.compstruc.2015.08.010}
}

@article{Surrogates1,
title = {Reduced order models for random functions. Application to stochastic problems},
journal = {Applied Mathematical Modelling},
volume = {33},
number = {1},
pages = {161-175},
year = {2009},
issn = {0307-904X},
doi = {https://doi.org/10.1016/j.apm.2007.10.023},
url = {https://www.sciencedirect.com/science/article/pii/S0307904X07002818},
author = {M. Grigoriu},
keywords = {Hydraulic head, Modal frequencies, Optimization, Pattern classification, Reduced order models, Stochastic equations},
abstract = {A method is developed for constructing reduced order models for arbitrary random functions. The reduced order models are simple random functions, that is, functions with a finite range (x1,…,xm). The construction of the reduced order models involves two steps. First, a range (x1,…,xm) is selected based on somewhat heuristic arguments. Second, the probabilities (p1,…,pm) of (x1,…,xm) are obtained from the solution of an optimization problem. Reduced order models are applied to calculate the distributions of the modal frequencies of a linear dynamic system with random stiffness matrix and statistics of the hydraulic head in a soil deposit with random heterogeneous conductivity. The performance of reduced order models in both applications is remarkable.}
}


@misc{PCE1,
      title={Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applications},
      author={Nora Lüthen and Stefano Marelli and Bruno Sudret},
      year={2021},
      eprint={2009.04800},
      archivePrefix={arXiv},
      primaryClass={stat.CO}
}

@Article{Nataf1,
 Author = {A. {Nataf}},
 Title = {{D\'etermination des distributions de probabilit\'es dont les marges sont donn\'ees}},
 FJournal = {{Comptes Rendus Hebdomadaires des S\'eances de l'Acad\'emie des Sciences, Paris}},
 Journal = {{C. R. Acad. Sci., Paris}},
 ISSN = {0001-4036},
 Volume = {255},
 Pages = {42--43},
 Year = {1962},
 Publisher = {Gauthier-Villars, Paris},
 Language = {French},
 Zbl = {0109.11904}
}

@article{Nataf2,
title = {An innovating analysis of the Nataf transformation from the copula viewpoint},
journal = {Probabilistic Engineering Mechanics},
volume = {24},
number = {3},
pages = {312-320},
year = {2009},
issn = {0266-8920},
doi = {https://doi.org/10.1016/j.probengmech.2008.08.001},
url = {https://www.sciencedirect.com/science/article/pii/S0266892008000660},
author = {Régis Lebrun and Anne Dutfoy},
keywords = {, Copula, Stochastic modelling},
abstract = {This article gives new insight on the Nataf transformation, a widely used tool in reliability analysis. After recalling some basics concerning the copula theory, we explain this transformation in the light of the copula theory and we uncover all the hidden hypothesis made on the dependence structure of the probabilistic model when using this transformation. Some important results concerning dependence modelling are given, such as the risk related to the use of a linear correlation matrix to describe the dependence structure, and the importance of tail dependence in probabilistic modelling for safety assessment. This contribution should allow the reader to be much more aware of the pitfalls in dependence modelling when relying solely on the Nataf transformation.}
}

@article{Utilities1,
title = {Computing a nearest symmetric positive semidefinite matrix},
journal = {Linear Algebra and its Applications},
volume = {103},
pages = {103-118},
year = {1988},
issn = {0024-3795},
doi = {https://doi.org/10.1016/0024-3795(88)90223-6},
url = {https://www.sciencedirect.com/science/article/pii/0024379588902236},
author = {Nicholas J. Higham},
abstract = {The nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + AT)/2. In the 2-norm a nearest symmetric positive semidefinite matrix, and its distance δ2(A) from A, are given by a computationally challenging formula due to Halmos. We show how the bisection method can be applied to this formula to compute upper and lower bounds for δ2(A) differing by no more than a given amount. A key ingredient is a stable and efficient test for positive definiteness, based on an attempted Choleski decomposition. For accurate computation of δ2(A) we formulate the problem as one of zero finding and apply a hybrid Newton-bisection algorithm. Some numerical difficulties are discussed and illustrated by example.}
}

@misc{nearest_psd,
       title={NEARESTSPD},
       url={https://www.mathworks.com/matlabcentral/fileexchange/42885-nearestspd},
       journal={MathWorks},
       author={D'Errico, John}
}

@article{Utilities3,
  doi = {10.1137/050624509},
  url = {https://doi.org/10.1137/050624509},
  year = {2006},
  month = jan,
  publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
  volume = {28},
  number = {2},
  pages = {360--385},
  author = {Houduo Qi and Defeng Sun},
  title = {A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix},
  journal = {{SIAM} Journal on Matrix Analysis and Applications}
}

@article{dsilva2018parsimonious,
  title={Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study},
  author={Dsilva, Carmeline J and Talmon, Ronen and Coifman, Ronald R and Kevrekidis, Ioannis G},
  journal={Applied and Computational Harmonic Analysis},
  volume={44},
  number={3},
  pages={759--773},
  year={2018},
  publisher={Elsevier}
}

################ Sensitivity Analysis ########################

# Morris
@article{Morris1,
title = {An effective screening design for sensitivity analysis of large models},
journal = {Environmental Modelling & Software},
volume = {22},
number = {10},
pages = {1509-1518},
year = {2007},
note = {Modelling, computer-assisted simulations, and mapping of dangerous phenomena for hazard assessment},
issn = {1364-8152},
doi = {https://doi.org/10.1016/j.envsoft.2006.10.004},
url = {https://www.sciencedirect.com/science/article/pii/S1364815206002805},
author = {Francesca Campolongo and Jessica Cariboni and Andrea Saltelli},
keywords = {Sensitivity analysis, Screening problem, Model-free methods, Effective sampling strategy, Dimethylsulphide (DMS)},
abstract = {In 1991 Morris proposed an effective screening sensitivity measure to identify the few important factors in models with many factors. The method is based on computing for each input a number of incremental ratios, namely elementary effects, which are then averaged to assess the overall importance of the input. Despite its value, the method is still rarely used and instead local analyses varying one factor at a time around a baseline point are usually employed. In this piece of work we propose a revised version of the elementary effects method, improved in terms of both the definition of the measure and the sampling strategy. In the present form the method shares many of the positive qualities of the variance-based techniques, having the advantage of a lower computational cost, as demonstrated by the analytical examples. The method is employed to assess the sensitivity of a chemical reaction model for dimethylsulphide (DMS), a gas involved in climate change. Results of the sensitivity analysis open up the ground for model reconsideration: some model components may need a more thorough modelling effort while some others may need to be simplified.}
}

# Chatterjee
@article{Chatterjee,
author = {Sourav Chatterjee},
title = {A New Coefficient of Correlation},
journal = {Journal of the American Statistical Association},
volume = {116},
number = {536},
pages = {2009-2022},
year  = {2021},
publisher = {Taylor & Francis},
doi = {10.1080/01621459.2020.1758115},
URL = {https://doi.org/10.1080/01621459.2020.1758115},
eprint = {https://doi.org/10.1080/01621459.2020.1758115}
}

@misc{gamboa2020global,
      title={Global Sensitivity Analysis: a new generation of mighty estimators based on rank statistics},
      author={Fabrice Gamboa and Pierre Gremaud and Thierry Klein and Agnès Lagnoux},
      year={2020},
      eprint={2003.01772},
      archivePrefix={arXiv},
      primaryClass={math.ST}
}

# Cramér-von Mises index
@article{CVM,
author = {Gamboa, Fabrice and Klein, Thierry and Lagnoux, Agnès},
title = {Sensitivity Analysis Based on Cramér--von Mises Distance},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
volume = {6},
number = {2},
pages = {522-548},
year = {2018},
doi = {10.1137/15M1025621},
URL = {https://doi.org/10.1137/15M1025621},
eprint = {https://doi.org/10.1137/15M1025621},
}


# Generalised Sobol index
@article{GSI,
author = {Fabrice Gamboa and Alexandre Janon and Thierry Klein and Agnès Lagnoux},
title = {{Sensitivity analysis for multidimensional and functional outputs}},
volume = {8},
journal = {Electronic Journal of Statistics},
number = {1},
publisher = {Institute of Mathematical Statistics and Bernoulli Society},
pages = {575 -- 603},
keywords = {Concentration inequalities, quadratic functionals, Semi-parametric efficient estimation, sensitivity analysis, Sobol indices, temporal output, vector output},
year = {2014},
doi = {10.1214/14-EJS895},
URL = {https://doi.org/10.1214/14-EJS895}
}

# Sobol
@book{saltelli_2008,
  author = {Saltelli, A.},
  description = {Global sensitivity analysis: the primer - Andrea Saltelli},
  isbn = {9780470059975},
  keywords = {sensitivity statistics},
  lccn = {2007045551},
  publisher = {John Wiley},
  title = {Global sensitivity analysis: the primer},
  url = {https://onlinelibrary.wiley.com/doi/book/10.1002/9780470725184},
  year = 2008
}

@article{saltelli_2002,
title = {Making best use of model evaluations to compute sensitivity indices},
journal = {Computer Physics Communications},
volume = {145},
number = {2},
pages = {280-297},
year = {2002},
issn = {0010-4655},
doi = {https://doi.org/10.1016/S0010-4655(02)00280-1},
url = {https://www.sciencedirect.com/science/article/pii/S0010465502002801},
author = {Andrea Saltelli},
keywords = {Sensitivity analysis, Sensitivity measures, Sensitivity indices, Importance measures},
}

@Article{PTMCMC1,
author ="Earl, David J. and Deem, Michael W.",
title  ="Parallel tempering: Theory{,} applications{,} and new perspectives",
journal  ="Phys. Chem. Chem. Phys.",
year  ="2005",
volume  ="7",
issue  ="23",
pages  ="3910-3916",
publisher  ="The Royal Society of Chemistry",
doi  ="10.1039/B509983H",
url  ="http://dx.doi.org/10.1039/B509983H",
abstract  ="We review the history of the parallel tempering simulation method. From its origins in data analysis{,} the parallel tempering method has become a standard workhorse of physicochemical simulations. We discuss the theory behind the method and its various generalizations. We mention a selected set of the many applications that have become possible with the introduction of parallel tempering{,} and we suggest several promising avenues for future research."}


@inproceedings{PTMCMC2,
title = {Using Thermodynamic Integration to Calculate the Posterior Probability in Bayesian Model Selection Problems},
booktitle = {AIP Conference Proceedings 707, 59},
year = {2004},
doi = {https://doi.org/10.1063/1.1751356},
author = {Paul M. Goggans and Ying Chi}
}


@article{STMCMC_ChingChen,
author = {Jianye Ching  and Yi-Chu Chen },
title = {Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging},
journal = {Journal of Engineering Mechanics},
volume = {133},
number = {7},
pages = {816-832},
year = {2007},
doi = {10.1061/(ASCE)0733-9399(2007)133:7(816)},
URL = {https://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28816%29},
eprint = {https://ascelibrary.org/doi/pdf/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28816%29}
}



@article{Kle2D,
title = {Simulation of multi-dimensional random fields by Karhunen–Loève expansion},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {324},
pages = {221-247},
year = {2017},
issn = {0045-7825},
doi = {https://doi.org/10.1016/j.cma.2017.05.022},
url = {https://www.sciencedirect.com/science/article/pii/S0045782516318692},
author = {Zhibao Zheng and Hongzhe Dai},
keywords = {Multi-dimensional random field, Karhunen–Loève expansion, Random field simulation, Fredholm integral equation},
}

@misc{FORM_XDu,
  title     = "Probabilistic Engineering Design, Chapter 7, First Order and Second Reliability Methods",
  author    = "Xiaoping Du",
  year      = 2005,
  publisher   = "University of Missouri – Rolla",
  howpublished = "\url{https://pdesign.sitehost.iu.edu/me360/ch7.pdf}"
}
